Problem: Determine how many solutions exist for the system of equations. ${6x-3y = -27}$ ${y = 2x+6}$
Solution: Convert both equations to slope-intercept form: ${6x-3y = -27}$ $6x{-6x} - 3y = -27{-6x}$ $-3y = -27-6x$ $y = 9+2x$ ${y = 2x+9}$ ${y = 2x+6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x+9}$ ${y = 2x+6}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.